Weight is not always an issue. It's a personal preference. It's a major stigma in our society but there are many people to whom weight is not an issue. There are other things so much more important than weight: personality, hygiene, motivation, sense of humour, etc, which of course are all subjective as well. Each person has different preferences. I would discard what that coach told you about it always being an issue. It's not true. When I look around, I see plenty of "unattractive" people getting dates, because they're confident. And to most people that's the biggest turn-on.
I can understand feeling weird about not having a date/relationship til this far in your life but you are actually still young, and (just my opinion) there's nothing wrong with that at all. Everyone has a different path in life. I guess it's just important that you identify what it is that you want. Dates? You want to meet women? Many people have a difficult time meeting and connecting with others. And not just on a dating level. If you say that you've made female friends then that is a good sign; it means you know how to communicate with women and that you're not just starting from the ground up, so to speak.
I agree with zinco. If you meet someone you're interested in, just ask her to go for coffee. It doesn't have to be a big deal (although I'm sure it will be, to you), just coffee for starters. Get to know her. You may go on tons of coffees before meeting someone you have a deeper connection with. You may end up going on many dates with women. And once you start it won't be a big deal anymore.
I think the important thing to remember is that everyone's journey is different, and that there is no right or wrong way. (Unless, of course, it's harming someone.)